Science from An Old Bucket of Water

wallpaper-mountains-1366x768
Trees’s growth  ignore the slope. Picture Source: https://previews.123rf.com/images/waldru/waldru1303/waldru130300029/18594282-Trees-on-the-mountain-slope-Winter-without-snow-Stock-Photo.jpg

If you climb a hill, you will notice that the trees are not perpendicular to the slope. With some variations due to wind, tree trunks generally meet the horizon at a right angle. Similarly, if you are collecting rain water with an old bucket on a slanted driveway, you will notice that the water level is not parallel to the asphalt. More water leans against the walls of the bucket facing downslope.

bucket

If you wait for more rain to fall into the bucket it will look like the second illustration. Wait longer and liquid water will not reach point C.  Instead it will overflow at point A. The bucket never fills with liquid.

All this is a reminder that gravity neither acts from the surface just below trees, nor from the surface below the driveway. It acts from the center of the Earth. We know from geometry that any radius is perpendicular to a line tangent to the circle. The horizon is equivalent to a tangent line, hence the reason for the alignment of trees and for the fact that points A and B in the bucket are at the same height above the horizon. (see green arrows in the diagram) It’s at those positions that they have the same potential energy, the product of weight and distance from the plane’s center.

If you leave the bucket out in late autumn, you will be in for a more pleasant surprise. After I knocked the bucket down, here’s what slid out along with the water that had not yet frozen.

icebucket
Two views of the same hollow cylinder of ice from a bucket. My keys are at the base for a sense of scale.

The ice is in the form of a hollow cylinder, one that eventually would have filled the whole bucket. The sight is a little deceiving. Unfrozen water is not the only thing missing from the picture. On its way out, the unfrozen water broke through a thin layer of surface-ice, which formed first because it’s the only part of the water that was directly in contact with the cold air. But why was the core of the liquid left unfrozen? The inner plastic walls not only cooled off faster than the subsurface water due to the lower specific heat of plastic, but the impurities and imperfections on its surface also provided nucleation sites for ice crystals to form. Even at the very beginning, one observes a crescent of thin ice on the colder surface, not coincidentally resting on the side of the bucket with more plastic exposed.

If we had waited long enough, why would the entire bucket have been filled with ice, something liquid water is incapable of doing when the bucket is sitting on a slope? On average, in liquid water, each molecule is hydrogen-bonded to about 3.4 neighboring molecules that constantly break and reform. But in ice each each molecule is hydrogen-bonded to 4 other molecules in a more stable fashion. This spreads out the H2O molecules in the ice structure, lowering its density. It’s the reason ice expands as it freezes into a hexagonal network, one that’s 3 kJ/mole stronger than that of a non-supercooled liquid network. It’s also the reason ice can’t flow like liquid water.

ice2
Text and image modified from a combo of AP Biology and chemguide.co.uk diagrams.

So after the ice starts to form on top and then along the internal walls of the bucket, the frozen base and the circular rim begin to thicken. The rest of the forming crystals grow until they reach the middle of the bucket. With molecules that are more tightly bound, the ice at point A cannot flow out of the bucket as it did when it was in a liquid state. But the air space in the bucket above the slant will be occupied as the ice from the freezing core expands and pushes upwards. The bucket, as a result, even though on a slant, gets completely filled with ice, which also expands against the bottom, deforming the plastic base.

If you slide the ice back into the bucket and wait for a warm day to melt it, water will reveal its intrinsic color. Most glass containers aren’t large enough for water to absorb enough red light to reveal a perceptible hue of greenish blue. Too often we get the false impression that water is transparent. But the white walls of the bucket will provide enough internal reflection to increase the path length, and water’s color becomes noticeable. I’ve ventured a little deeper into that idea in this blog entry. If you’re not interested, hopefully I’ve nevertheless shown that there are some side-benefits to saving water and energy while collecting rain.

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Specific Heat: a Beautiful Characteristic Property

More than once, I intended to prepare  hard boiled eggs  for my kids in the morning, placed water in a pot but forgot the egg. If I remembered within 30 seconds or so, it was still safe to place my hand in the water, but it was  a bad idea to touch the pot itself. Metals warm up faster than water does—that’s what a low specific heat implies. Specific heat is a characteristic property that measures how much energy it takes to raise the temperature of 1 gram of a material by 1 oC.  The higher the c value is, the more difficult it is to warm up that substance. By the same token, substances with high specific heats also lose their heat with difficulty, while metals cool off with ease.

The high specific heat capacity of water helps temper the rate at which air changes temperature, which is why temperature changes between seasons is gradual, especially near large lakes or the ocean. Water is the reason why Toronto is milder than Montreal.

from http://water.usgs.gov/edu/heat-capacity.html
from http://water.usgs.gov/edu/heat-capacity.html

It’s the reason that the Gulf stream can retain the heat of water heated in the Caribbean and carry it towards London, England,  a city with an average temperature of 7 ºC, even though its latitude is six degrees more North than that of Montreal, whose average January- temperature is only -9 ºC.

If water’s specific heat was lower than it actually is, life would not be possible. The rate of evaporation would be too high, and it would be too difficult for the evolutionary precursors of cells to maintain homeostasis. Why is water special? The reason it is difficult to raise water’s temperature is that hydrogen bonds exist between molecules of water. The hydrogen atoms of one molecule are attracted to the oxygen atoms of other molecules. (see the dots = . . . . in the adjacent diagram. )

Hbonds
Five molecules linked by hydrogen bonds, represented by the four-dot sequence.

Each set represents a hydrogen bond. Five molecules are shown in all)) To overcome this attraction, energy is needed. The bonds between the water molecules are like the links between the wagons of a train. Just like it is difficult to get a big train to reach a high speed, it takes a lot of energy to warm up water. Once the train is moving, it is difficult to stop. Similarly it is difficult to cool water. Of course, in the liquid state, the links are not as fixed as those of train links. As molecules rotate, they constantly break and reform H-bonds, but their overall effect is to maintain attractions within the group.

A very useful formula allows us to calculate the amount of heat either absorbed or lost by a substance during a physical change: Q = m c ΔT , where Q = quantity of heat,  m = mass being heated or cooled, c is the specific heat and  ΔT is the change in temperature.

Now imagine two identical masses, of say, water and copper,  each at the same temperature, receiving the same amount of heat. Since Q is equal, then  

 the masses in the equation cancel, and we end up with:

spHeat

The inverse relationship graphically represented within the cross reveals that for water’s product cwΔTw to be equal to that of copper, a bigger c  entails a smaller ΔT. For copper it’s the reverse. A smaller c entails a bigger ΔT.

While water’s specific heat of 4.19 J/(g º C) is elevated to that figure by its intermolecular bonds, no such attractions exist between copper atoms. If temperature extremes are excluded, the specific heat of most metals can be predicted from c = 3 R/M, where R = gas constant 8.31 J/mole K and M = molar mass in grams per mole. Since c, in our context , is based on a temperature difference, any  ΔT is equivalent in both C(Celsius) or K(Kelvin) scale of temperature.  c = 3*8.31 J/mole K / (63.5 g/mole)= 0.392 J/(g K )= 0.392 J/(g º C), close to the experimental value of 0.386 obtained by dropping a heated, weighed piece of pure copper into the cool water of a calorimeter and measuring the maximum temperature attained by the water. ( Both the initial hot and cool temperatures also have to be measured).

 

 

 

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